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The classical theta correspondence, based on the Weil representation, allows one to lift automorphic representations on symplectic groups or their double covers to au- tomorphic representations on special orthogonal groups. It is of interest to vary the orthog- onal group and describe the behavior in this theta tower (the Rallis tower). In prior work, the authors obtained an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups that is based on the tensor product of the Weil representation with another small representation. In this work we study the existence of generic lifts in the resulting theta tower. In the classical case, there are two orthogonal groups that may support a generic lift of an irreducible cuspidal automorphic representation of a symplectic group. We show that in general the Whittaker range consists of r + 1 groups for the lift from the r-fold cover of a symplectic group. We also give a period criterion for the genericity of the lift at each step of the tower. 

The classical theta correspondence establishes a relationship between automorphic representations on special orthogonal groups and automorphic representations on symplectic groups or their double covers. This correspondence is achieved by using as integral kernel a theta series on the metaplectic double cover of a symplectic group that is constructed from the Weil representation. There is also an analogous local correspondence. In this work we present an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups. The key issue here is that for higher degree covers there is no analogue of the Weil representation, and additional ingredients are needed. Our work reflects a broader paradigm: constructions in automorphic forms that work for algebraic groups or their double covers should often extend to higher degree metaplectic covers.